493 research outputs found
Tournament Sequences and Meeussen Sequences
A "tournament sequence" is an increasing sequence of positive integers
(t_1,t_2,...) such that t_1=1 and t_{i+1} <= 2 t_i. A "Meeussen sequence" is an
increasing sequence of positive integers (m_1,m_2,...) such that m_1=1, every
nonnegative integer is the sum of a subset of the {m_i}, and each integer m_i-1
is the sum of a unique such subset.
We show that these two properties are isomorphic. That is, we present a
bijection between tournament and Meeussen sequences which respects the natural
tree structure on each set. We also present an efficient technique for counting
the number of tournament sequences of length n, and discuss the asymptotic
growth of this number. The counting technique we introduce is suitable for
application to other well-behaved counting problems of the same sort where a
closed form or generating function cannot be found.Comment: 16 pages, 1 figure. Minor changes only; final version as published in
EJ
Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Pl\"ucker relations
We present a ``method'' for bijective proofs for determinant identities,
which is based on translating determinants to Schur functions by the
Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a
bijective construction (which was first used by Goulden) to a class of Schur
function identities, from which we shall obtain bijective proofs for Dodgson's
condensation formula, Pl\"ucker relations and a recent identity of the second
author.Comment: Co-author Michael Kleber added a new proof of his theorem by
inclusion-exclusio
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